Title: TQFTs through Parallel Transport and Quantum Computing
Abstract: Quantum computing is captured in the formalism of the monoidal subcategory of Vect (category of complex vector spaces) generated by C^2 – in particular, quantum circuits are diagrams in Vect – while topological quantum field theories, in the sense of Atiyah, are diagrams in Vect indexed by cobordisms. We outline a program to formalize this connection. In doing so, we first equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a double category. Finite dimensional complex vector spaces and linear maps between them are given a suitable double categorical structure which we call FVect. Finally, we realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to FVect, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category. This talk reports on joint work with Steven Rayan.